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38536 results in Cambridge Textbooks

Page 1172 of 1927

  • 15 - Critical stress state analysis of beams

  • from Part 2 - Prestressed concrete
  • Yew-Chaye Loo, Griffith University, Queensland, Sanaul Huq Chowdhury, Griffith University, Queensland
  • Book:
    Reinforced and Prestressed Concrete
    Published online:
    12 December 2018
    Print publication:
    26 July 2018, pp 414-436

  • Summary

    Introduction

    By definition, a fully prestressed beam sustains neither tensile cracking nor overstress in compression, under any given service load. Achieving these no-crack and no-overstress conditions throughout the working life of a beam – when prestress losses occur instantaneously and continuously – is a complicated problem. The critical stress state (CSS) approach presented in this chapter provides a fool-proof solution to this otherwise intractable problem. It is a linear–elastic method and is valid subject to the following assumptions:

  • • The plane section remains plane after bending.

  • • The material behaves elastically.

  • • The beam section is homogenous and uncracked.

  • • The principle of superposition holds.

    • Note that the CSS approach is suitable for partially prestressed beams sustaining tensile stresses below the concrete cracking strength (see Section 14.5).

    Notation

    Figure 15.2(1)a illustrates a typical section of a prestressed I-shaped bridge beam. It may be idealised as shown in Figure 15.2(1)b in which the resultant H of the individual prestressing forces is located at the effective centre of prestress, or with an eccentricity (eB) from the neutral axis (NA). The effective centre is the centre of gravity (action) of the individual prestressing forces. Its location, or the value of eB, can be determined by simple statics.

    To develop the CSS equations in a systematic manner, specific notation needs to be followed. As shown in Figure 15.2(1)b, the eccentricity (e) is positive upwards and negative downwards. Consequently, the moment with respect to NA created by H is –HeB. The sign convention for the beam moment is given in Figure 15.2(2): positive (+) where it produces compression above NA, and negative (–) for tension below it.

    The concrete stress fc is positive (+) for compression and negative (–) for tension. Note that this convention may be opposite to that adopted in some books on the mechanics of materials.

    For the section shown in Figure 15.2(1), under a working load moment (Mw), we can write, by virtue of the principle of superposition, the following equations:

  • 17 - Ultimate strength analysis of beams

  • from Part 2 - Prestressed concrete
  • Yew-Chaye Loo, Griffith University, Queensland, Sanaul Huq Chowdhury, Griffith University, Queensland
  • Book:
    Reinforced and Prestressed Concrete
    Published online:
    12 December 2018
    Print publication:
    26 July 2018, pp 458-472

  • Summary

    Introduction

    As stated in previous chapters, the critical stress state (CSS) design method is based on linear–elastic or permissible-stress principles. The beam design is considered to comply with AS 3600–2009 (the Standard) requirements only if its strength satisfies

    Before proceeding to develop the formulas for Mu, the following observations are made:

  • • The behaviour of prestressed beam at ultimate is similar to its reinforced counterpart.

  • • Assumptions made for reinforced beams in Section 3.3.1 are also valid for prestressed beams.

  • • The stress–strain curves and allied properties of prestressing and reinforcing steel are as given in Figure 17.1(1).

  • • The rectangular stress block as defined in Figure 3.3(3) is a valid equivalent to the actual stress distribution of concrete at ultimate.

  • • A prestressed section fails when εc = 0.003.

    • Cracking moment (MCR)

    Formula

    For the prestressed beam shown in Figure 17.2(1), the beam would crack if the bottom-fibre tensile stress reached the flexural tensile strength of concrete.

    Based on Equation 15.3(8), and under the prestress force H at eB from the neutral axis (NA), the compressive bottom-fibre stress after loss may be written as

    Thus, the cracking moment

    where as per Equation 2.2(2) and η is the effective prestress coefficient.

    Illustrative example

    To demonstrate the application of Equation 17.2(2), an example is given below.

    Using the information given for the beam detailed in Section 15.8, what is Mcr, assuming

    Solution

    From Section 15.8, we know:

  • • ZB = 20.68 × 108/250 = 8.272 × l06 mm3

  • • ηH = 0.85 × 667500 = 567375 N

  • • eB = 175.5 mm

  • • A = 72.5 × l03 mm2

  • 13 - Strut-and-tie modelling of concrete structures

  • from Part 1 - Reinforced concrete
  • Yew-Chaye Loo, Griffith University, Queensland, Sanaul Huq Chowdhury, Griffith University, Queensland
  • Book:
    Reinforced and Prestressed Concrete
    Published online:
    12 December 2018
    Print publication:
    26 July 2018, pp 377-400

  • Summary

    Introduction

    ‘Strut-and-tie modelling’ is described in Section 7 of AS 3600–2009 (the Standard). Since the original issue of AS 3600 (1988), the current Standard is the first update in which a separate section is devoted to this type of modelling. This chapter serves to complement Section 7 of the Standard.

    Figure 13.l(1) shows some typical reinforced and prestressed concrete structures and elements. In terms of stress distribution characteristics in response to external loads, each structure or element can be divided into the so-called ‘B’ and ‘D’ regions (Schlaich, Schafer and Jennewein 1987; Schlaich and Schafer 1991). In general, B regions are dominated by bending and D regions (the ‘distributed’ or discontinuous regions) are dominated by nonflexural stresses.

    The analysis and design of various B regions have been extensively explored in most chapters of this book. That the strength behaviour of B (bending) regions can be accurately determined or designed for, using explicit formulas or well-prescribed analytical procedures, is beyond doubt. However, the same cannot be said of the behaviour of the D regions. For these, the analysis or design in general requires rather crude empirical formulas or, alternatively, the aid of sophisticated computer-based numerical procedures, such as the finite element method. The strut-and-tie modelling technique, on the other hand, can provide a direct design process for many types of D region.

    The fundamental concepts of the strut-and-tie model (STM) and some recent developments of the technique are discussed in this chapter, typical models taking the form of statically determinate truss systems are illustrated, and the Standard's specifications for various types of struts, ties and ‘nodes’ are also described.

    Fundamentals

    A typical deep beam under a single concentrated load (P) at midspan is illustrated in Figure 13.2(1)a. The zones between the external load and the two reactions may be taken as a D region. Under the load P and the reactions P/2, the compressive and tensile stress distributions, as represented by the stress trajectories, are presented schematically in Figure 13.2(1)a. It is obvious that zones a–d–b–a and a–e–c–a are dominated by compressive stresses, whereas zone f–g–h is dominated by tensile stresses. As the span–depth ratio L/h becomes smaller, the compressive and tensile stress fields (zones) would become more and more distinctive.

  • 18 - End blocks for prestressing anchorages

  • from Part 2 - Prestressed concrete
  • Yew-Chaye Loo, Griffith University, Queensland, Sanaul Huq Chowdhury, Griffith University, Queensland
  • Book:
    Reinforced and Prestressed Concrete
    Published online:
    12 December 2018
    Print publication:
    26 July 2018, pp 473-482

  • Summary

    Introduction

    It is recognised in Section 14.1 that prestressing tendons (either in the form of wire or strands of wire) reinforces the weaknesses of concrete in an active manner. Because of this, considerable concentrated forces are exerted at the extremities of a prestressed beam. At the end zones, these forces in pretensioned beams translate into intensive bond stresses in the steel concrete interface. In post-tensioned beams, they induce acute lateral tensile stresses and the anchor heads (see Figure 14.4(3)a) create high bearing stresses on the concrete ends.

    These stresses need to be fully considered and carefully designed for, to prevent cracking and even premature failure in the end zones. A properly reinforced end zone is referred to as an end block.

    The nature and distribution of the bond stress in the end zones of a pretensioned beam are given in Section 18.2, which also includes the design method recommended in AS 3600–2009 (the Standard). Section 18.3 identifies the three types of stresses induced by a post-tensioned anchorage system. Two of these are the bursting stress and the spalling stress, both of which are tensile, and orthogonal or transverse in direction to the axis of the post-tensioned tendon. There is also the bearing (compressive) stress on the concrete right behind the steel anchor head. The design for the bursting, spalling and bearing stresses is discussed in Section 18.4. Finally, the distribution and detailing of the end-block reinforcement are presented in Section 18.5.

    Pretensioned beams

    For a pretensioning system, the transfer of the prestressing forces to concrete is achieved mainly by bond along the ‘transmission length’. Figure 18.2(1)a shows a tendon in the end zone of a pretensioned beam before transfer. Note in Figure 18.2(1)b that, at and after transfer, the tendon diameter (R) at the free end is greater than the embedded diameter (r). This is a result of Poisson's effect.

    The distribution of the resulting bond stress between tendon and concrete is illustrated in Figure 18.2(1)c. Figure 18.2(1)d depicts the gradually increasing tendon stress, which corresponds to the increase and decay of the bond stress.

  • 16 - Critical stress state design of beams

  • from Part 2 - Prestressed concrete
  • Yew-Chaye Loo, Griffith University, Queensland, Sanaul Huq Chowdhury, Griffith University, Queensland
  • Book:
    Reinforced and Prestressed Concrete
    Published online:
    12 December 2018
    Print publication:
    26 July 2018, pp 437-457

  • Summary

    Introduction

    In the bending design of prestressed members in general, and of beams in particular, the process below should be followed:

  • • Critical stress state (CSS) criteria must be satisfied at all stages of the life of the beam (i.e. at transfer, during handling, during construction, under service load conditions and after losses have occurred).

  • • If applicable, shear and torsion must be designed for and the CSS design modified if necessary (see Clauses 8.2–8.4 of AS 3600–2009 [the Standard]).

  • • The design must be checked for adequacy under ultimate load conditions (see Clause 8.1 of the Standard).

  • • End-block stresses must be estimated, and reinforcement provided (Clause 12.5 of the Standard).

  • • Deflections must be assessed and kept within acceptable limits (see Clause 8.5 of the Standard).

    • There are other general design requirements that have to be met, including durability (Section 4 of the Standard), fire resistance (Section 5), material properties (Section 3), and other serviceability considerations including crack control (for partially prestressed beams only) and vibration (Clause 9.5). For cracked partially prestressed beams, Equations 5.7(1) and (2) respectively may be used to estimate the average and maximum crack widths.

    This chapter mainly presents the CSS approach to bending design or how the first item in the above bulleted list is satisfied. Chapter 17 covers, in some detail, the ultimate strength check for fully and partially prestressed beams (the third bullet point above). Chapter 18 presents the end-block design for prestressing anchorages. The reader is referred to the Standard for details of the other design considerations listed above. It is worth noting that design topics such as shear, torsion, durability, material properties and crack control for reinforced beams have been discussed in detail in Part 1. The design procedures are similar, if not identical, for prestressed beams. Equations for shear and torsion design of reinforced beams, as well as deflection calculations, are truncated from the unified formulas given in the Standard for reinforced and prestressed beams. As durability and fire resistance requirements affect mainly detailing including concrete cover for steel, the Standard provisions are the same for the two types of beams.

  • 15 - Valuing Impacts from Observed Behavior: Indirect Market Methods

  • Anthony E. Boardman, University of British Columbia, Vancouver, David H. Greenberg, University of Maryland, Baltimore County, Aidan R. Vining, Simon Fraser University, British Columbia, David L. Weimer, University of Wisconsin, Madison
  • Book:
    Cost-Benefit Analysis
    Published online:
    12 December 2018
    Print publication:
    19 July 2018, pp 389-421

  • Summary

    Generally, estimation of changes in social surplus requires knowledge of entire demand and supply schedules. Chapter 4 discusses direct estimation of demand and supply curves, focusing on the demand curve for the purpose of measuring consumer surplus. It assumes that there is a market demand schedule for the good in question, such as garbage collection or gasoline, and we can observe at least one point on this demand curve. In many applications of CBA, however, the markets for certain “goods,” such as human life or pollution, do not exist or are imperfect for reasons discussed in Chapter 3.

  • 1 - Introduction to Cost–Benefit Analysis

  • Anthony E. Boardman, University of British Columbia, Vancouver, David H. Greenberg, University of Maryland, Baltimore County, Aidan R. Vining, Simon Fraser University, British Columbia, David L. Weimer, University of Wisconsin, Madison
  • Book:
    Cost-Benefit Analysis
    Published online:
    12 December 2018
    Print publication:
    19 July 2018, pp 1-27

  • Summary

    In the Affair of so much Importance to you, wherein you ask my Advice, I cannot for want of sufficient Premises, advise you what to determine, but if you please I will tell you how. When those difficult Cases occur, they are difficult, chiefly because while we have them under Consideration, all the Reasons pro and con are not present to the Mind at the same time; but sometimes one Set present themselves, and at other times another, the first being out of Sight. Hence the various Purposes or Inclinations that alternately prevail, and the Uncertainty that perplexes us.

  • Case 9 - A CBA of the North-East Mine Development Project

  • Anthony E. Boardman, University of British Columbia, Vancouver, David H. Greenberg, University of Maryland, Baltimore County, Aidan R. Vining, Simon Fraser University, British Columbia, David L. Weimer, University of Wisconsin, Madison
  • Book:
    Cost-Benefit Analysis
    Published online:
    12 December 2018
    Print publication:
    19 July 2018, pp 233-236

  • Summary

    The prices of most goods and services tend to rise over time, that is, we experience inflation. However, in practice, not all prices (or values) increase at the same rate. Some prices, such as house prices and fuel prices, are often much more volatile than others. For this reason, some countries exclude such volatile items from their basket of goods when computing the CPI. And the prices of some goods and services sometimes go in a different direction to other prices. For example, from December 2010 to December 2016 in the US, the all-items CPI rose about 10 percent; however, the price of houses rose about 21 percent, while the price of gold fell about 15 percent.1

Page 1172 of 1927